Vectors & 3D Geometry15 questions15 PYQ

Vectors & 3D Geometry — JEE Maths Practice Questions & Solutions

15 questions on Vectors & 3D Geometry with full step-by-step solutions, including past-year (PYQ) problems. Free to practice.

mediumPYQ · JEE Main 2026
If the distance of the point (a,2,5)(a,2,5) from the image of (1,2,7)(1,2,7) in the line
x1=y−11=z−22\dfrac{x}{1}=\dfrac{y-1}{1}=\dfrac{z-2}{2}
is 4, then the sum of all possible values of aa is:
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mediumPYQ · JEE Main 2026
A line with direction ratios 1,−1,21,-1,2 intersects the lines x2=y3=z+13\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z+1}{3} and x+1−1=y−21=z4\dfrac{x+1}{-1}=\dfrac{y-2}{1}=\dfrac{z}{4} at the points PP and QQ respectively. If the length of the line segment PQPQ is α\alpha, then 225α2225\alpha^2 is equal to
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mediumPYQ · JEE Main 2026
Let OO be the origin with OP→=a⃗\overrightarrow{OP}=\vec{a} and OQ→=b⃗\overrightarrow{OQ}=\vec{b}. If RR is on OP→\overrightarrow{OP} such that OP→=5OR→\overrightarrow{OP}=5\overrightarrow{OR}, and MM is such that OQ→=5RM→\overrightarrow{OQ}=5\overrightarrow{RM}, then PM→\overrightarrow{PM} equals:
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mediumPYQ · JEE Main 2026
Let △PQR\triangle PQR be such that PP and QQ lie on x+38=y−42=z+12\dfrac{x+3}{8}=\dfrac{y-4}{2}=\dfrac{z+1}{2} and are at distance 6 from R(1,2,3)R(1,2,3). If (α,β,γ)(\alpha,\beta,\gamma) is the centroid of △PQR\triangle PQR, then α+β+γ\alpha+\beta+\gamma equals:
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mediumPYQ · JEE Main 2026
Let a⃗k=(tan⁡θk)i^+j^\vec a_k=(\tan\theta_k)\hat i+\hat j and b⃗k=i^−(cot⁡θk)j^\vec b_k=\hat i-(\cot\theta_k)\hat j, where θk=2k−1π2n+1\theta_k=\dfrac{2^{k-1}\pi}{2^n+1}, for some n∈N, n>5n\in\mathbb{N},\ n>5. Then the value of ∑k=1n∣a⃗k∣2∑k=1n∣b⃗k∣2\dfrac{\displaystyle\sum_{k=1}^{n}|\vec a_k|^2}{\displaystyle\sum_{k=1}^{n}|\vec b_k|^2} is
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mediumPYQ · JEE Main 2026
The square of the distance of the point P(5,6,7)P(5,6,7) from the line x−22=y−53=z−24\dfrac{x-2}{2}=\dfrac{y-5}{3}=\dfrac{z-2}{4} is
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mediumPYQ · JEE Main 2026
The square of the distance of the point of intersection of the lines r⃗=(i^+j^−k^)+λ(ai^−j^)\vec r=(\hat i+\hat j-\hat k)+\lambda(a\hat i-\hat j), a≠0a\ne0, and r⃗=(4i^−k^)+μ(2i^+ak^)\vec r=(4\hat i-\hat k)+\mu(2\hat i+a\hat k) from the origin is
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mediumPYQ · JEE Main 2026
Let a⃗=7 i^+j^−k^\vec a=\sqrt7\,\hat i+\hat j-\hat k and b⃗=j^+2k^\vec b=\hat j+2\hat k. If r⃗\vec r is a vector such that r⃗×a⃗+a⃗×b⃗=0⃗\vec r\times\vec a+\vec a\times\vec b=\vec 0 and r⃗⋅a⃗=0\vec r\cdot\vec a=0, then ∣3r⃗∣2|3\vec r|^2 is equal to
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mediumPYQ · JEE Main 2026
If (2α+1, α2−3α, α−12)\left(2\alpha+1,\ \alpha^2-3\alpha,\ \dfrac{\alpha-1}{2}\right) is the image of (α, 2α, 1)(\alpha,\ 2\alpha,\ 1) in the line x−23=y−12=z1\dfrac{x-2}{3}=\dfrac{y-1}{2}=\dfrac{z}{1}, then the possible value(s) of α\alpha is(are)
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mediumPYQ · JEE Main 2026
Let u^\hat u and v^\hat v be unit vectors inclined at an acute angle such that ∣u^×v^∣=32|\hat u\times\hat v|=\dfrac{\sqrt3}{2}. If A⃗=λu^+v^+(u^×v^)\vec A=\lambda\hat u+\hat v+(\hat u\times\hat v), then λ\lambda is equal to
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mediumPYQ · JEE Main 2026
Let a line LL passing through the point (1,1,1)(1,1,1) be perpendicular to both the vectors 2i^+2j^+k^2\hat i+2\hat j+\hat k and i^+2j^+k^\hat i+2\hat j+\hat k. If P(a,b,c)P(a,b,c) is the foot of perpendicular from the origin on the line LL, then the value of 34(a+b+c)34(a+b+c) is
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mediumPYQ · JEE Main 2026
If the point of intersection of the lines x+13=y+a5=z+b+17\dfrac{x+1}{3}=\dfrac{y+a}{5}=\dfrac{z+b+1}{7} and x−21=y−b4=z−2a7\dfrac{x-2}{1}=\dfrac{y-b}{4}=\dfrac{z-2a}{7} lies on the xyxy-plane, then the value of a+ba+b is
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easyPYQ · JEE Main 2026
The square of the distance of the point (−2,−8,6)(-2,-8,6) from the line x−11=y−12=z−1\dfrac{x-1}{1}=\dfrac{y-1}{2}=\dfrac{z}{-1} along the line x+51=y+5−1=z2\dfrac{x+5}{1}=\dfrac{y+5}{-1}=\dfrac{z}{2} is equal to
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easyPYQ · JEE Main 2026
The shortest distance between the lines r⃗=(13i^+2j^+83k^)+λ(2i^−5j^+6k^)\vec r=\left(\dfrac13\hat i+2\hat j+\dfrac83\hat k\right)+\lambda\left(2\hat i-5\hat j+6\hat k\right) and r⃗=(−23i^−13k^)+μ(j^−k^)\vec r=\left(-\dfrac23\hat i-\dfrac13\hat k\right)+\mu\left(\hat j-\hat k\right), λ,μ∈R\lambda,\mu\in\mathbb{R}, is
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easyPYQ · JEE Main 2026
If a⃗\vec a and b⃗\vec b are two vectors such that ∣a⃗∣=2|\vec a|=2 and ∣b⃗∣=3|\vec b|=3, then the maximum value of 3∣(3a⃗+2b⃗)∣+4∣(3a⃗−2b⃗)∣3\big|(3\vec a+2\vec b)\big|+4\big|(3\vec a-2\vec b)\big| is
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